The pseudo-random binary sequence was the first periodic, discrete-level signal used to measure the frequency response at a number of harmonic fre­t See the literature (1-55).

quencies simultaneously. In addition to numerous tests on nuclear reactors, this signal has been used in tests for distillation columns (5,19, 39), oil refineries (44,47), evaporators (19), heat exchangers (7), steam generators (21), diesel engines (43), machine tools (29), electric furnaces (36, 37), paper mills (45), and steel mills (45).

Two different types of PRBS signals, called the quadratic residue sequences and the maximal-length or m sequences, are available. Quadratic residue sequences with Z bits may be generated for Z = 4/c — 1 for integer values of к if Z is a prime number; m sequences with Z bits may be generated for Z = 2n — 1 for integer values of n. Since m sequences may be generated easily using a shift-register algorithm, only these will be considered here. A description of the method for generating m sequences appears later in this section.

The two main characteristics of interest for a PRBS are its power spectrum and its autocorrelation function. These are as follows.

(1) Power Spectrum

Pk

Pk

where Pk is the power in the fcth harmonic, A the signal amplitude (the input is A or — A), Z the number of bits, and к the harmonic number. For example, two periods of a 7-bit PRBS appear in Fig. 3.1. Power spectra for several

PRBS signals appear in Fig. 3.2. In order to make the cases comparable, all are based on the same period and the same input amplitude.

Harmonic Fig. 3.2. Power spectra for PRBS signals.

(2) Autocorrelation Function Cn(t) = 1 — [(Z + 1)/T]t for 0 < t < T/Z = -1 /Z for T/Z <, t < (Z — 1 )T/Z = — Z + [(Z + 1)/T]t for (Z — 1 )T/Z < T < T

where Cі i(t) is the autocorrelation function, т the lag time, and T the period. The general form of the autocorrelation appears in Fig. 3.3.

From these results we can deduce the following:

1. The autocorrelation function has a spike at 0, T, 2T,… and a negative bias of 1/Z between spikes. As the number of bits increases, the spike becomes sharper and the bias becomes smaller. Thus the autocorrelation function looks more like a series of delta functions as Z increases.

2. The power spectrum becomes flatter as Z increases. This may be useful in tests in which it is desired to measure the frequency response at a number of harmonic frequencies.

3. The absolute magnitude of the harmonics decreases as Z increases. This is expected, since the same total power is distributed rather evenly over more harmonics for the longer sequences. This suggests that a compromise may be required in selecting a sequence for a particular application. Long sequences give the desired flat spectra, but the signal-to-noise ratio at each harmonic may be too low because of inadequate signal strength.

We observe that the power spectrum of the PRBS is smaller than the maxi­mum possible power spectrum of a binary pulse chain (Eq. 2.12.7) by a constant factor, (Z + 1)/Z2. Thus the bandwidth relation derived in Section 2.12 applies for the PRBS. That is,

fc1/2 = 0.44Z (3.1.4)

where fc1/2 is the harmonic at which the signal power is down to half of its maximum value. A rule of thumb in estimating the useful frequency range of a signal is

2к/Т < a) < O. SSn/At	rad/sec	(3.1.5)
6.28/T < со < 2.76/At	rad/sec	(3.1.6)
1/T < / < 0.44/At	Hz	(3.1.7)

where At is the bit duration (in seconds). This only says that if there is enough energy in the first harmonic, then there is probably a good chance of obtain­ing results out to harmonic number 0.44Z because it has half as much energy as the first harmonic.

4. The PRBS is not antisymmetric. Because of this, the PRBS does not have the tendency to discriminate against nonlinear contamination possessed by the antisymmetric signals. (See Section 2.13.)

The pseudo-random binary sequence may be generated with a digital shift register with modulo-2 adder feedback. Modulo-p addition is defined for any two digits m and n, where m = 0,1,…, p — 1 and n = 0,1,…, p — 1. The modulo-p sum of m and n is the same as in common arithmetic if (m + n) < p. If (m — I — n) is equal to p, then the modulo-p sum is given by m + n — p. Thus modulo-2 addition is defined as shown in Table 3.1.

TABLE 3.1 Modulo-2 Addition m n + «)mod2 і 0 1 0 1 1 1 1 0 0 0 0

A three-stage digital shift register with feedback from stages 2 and 3 is shown in Fig. 3.4. The operation of this digital shift register with modulo-2

—H 11213

Fig. 3.4. A three-stage shift register with feedback.

adder feedback proceeds as follows:

1. Set initial values (0 or 1) on the stages of the digital shift register. They may be selected arbitrarily except for the requirement that at least one of them must be 1.

2. Generate a new set of digits for the stages of the shift register as follows.

a. Add modulo-2 the values in specified stages of the shift register.

b. Shift all existing values in the shift register one stage to the right and insert the feedback term from step (a) into stage 1. Note that the value in the last stage is removed when the shifting is accomplished.

3. Repeat until the initial values reappear in all of the stages of the shift register.

This set of operations produces a series of digits (0 or 1) for each stage in a pattern that repeats periodically. The pattern in each stage is the same, but the patterns are offset relative to one another. The 0 and 1 indicate which value of the two-level input signal to use. We may interpret either 0 or 1 as the high-level input or the low-level input. For instance, the three-bit sequence (1 0 1) may be experimentally implemented as (high, low, high) or (low, high, low). The correlation functions and power spectra are identical in each case.

It is known that the maximum number of bits in such a periodic sequence generated by an n-stage shift register with modulo-2 adder feedback is 2й — 1. Other shorter periodic sequences are obtained for certain feedback connections, but only the maximum-length sequence has the desired pseudo­random character. The arrangement of the feedback connections needed to generate maximum-length sequences are known for a number of sequences. In some cases, several different feedback connections will work. Some of the sequences may be generated with one modulo-2 adder and others require several. Table 3.2 gives feedback connections that require only one modulo-2 adder if a single adder connection is known. For others, connections are given for the two additions from the numbers in three stages.

TABLE 3.2 Feedback Connections for PRBS Signals n Z Stages to be added, modulo 2 2 3 1, 2 3 7 1, 3 4 15 1, 4 5 31 2, 5 6 63 1,6 7 127 1, 7 S 255 1, 2, 8 9 511 4, 9 10 1,023 3, 10 11 2,047 2, 11 12 .3,095 2, 7, 12 14 16,383 1, 2, 14 15 32,767 1, 15 16 67,535 1, 2, 16 17 131,071 3, 17 IS 262,143 7, 18 19 524,287 1, 7, 19 20 1,048,575 3, 20

As an example, consider the arrangement for a three-stage digital shift register shown in Fig. 3.5. Take the initial conditions on the three stages as 10 0. Then the following sequence is obtained.

1 0 0 Initial

1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1

1 0 0 Cycle repeats

Thus a 7-bit pseudo-random binary sequence is (1 1 1 0 1 00).

Two properties of pseudo-random binary sequences are:

(1) Zero Crossings. For a Z bit sequence, there are (Z + l)/2 zero crossings. This gives the number of shifts required of the input hardware per period and may be useful for assessing possible wear on this hardware during a test.

(2) Runs. Successive occurrence of one of the states is called a run. There are (Z + l)/2 runs per period. Half of the runs are of one-bit length; one — fourth are of two-bit length; one-eighth are of three-bit length; and so on, provided that the number of bits generated is greater than 1. In addition, there is one run of length n, where Z = 2" — 1.